Uniform boundedness and extinction results of solutions to a predator–prey system
Mokhtar Kirane
1, Salah Badraoui
B2and Mohammed Guedda
31LaSIE, Fac. des Sciences et Technologies, Univ. La Rochelle, 17042 La Rochelle, France
2Laboratory of Analysis and Control of Differential Equations “ACED”, Fac. MISM, Dept. Maths, Univ. 8 Mai 1945 Guelma, Algeria
3LAMFA, CNRS UMR 7352, Dept. Mathematics, Univ. Picardie Jules Verne, Amiens, France
Received 26 May 2019, appeared 3 February 2020 Communicated by Péter L. Simon
Abstract. Global existence, positivity, uniform boundedness and extinction results of solutions to a system of reaction-diffusion equations on unbounded domain modeling two species on a predator–prey relationship is considered.
Keywords: reaction-diffusion, global existence, positivity, predator–prey, extinction of solutions.
2010 Mathematics Subject Classification: 35A01, 35A02, 35B09, 35B50, 35K57.
1 Introduction
We consider the qualitative theory of the Cauchy problem for a system of reaction-diffusion equations modeling two species interacting with predator–prey relationship. The system in consideration is
La,ν ≡ut−auxx−νux =−pu+quv≡ f(u,v), x ∈R, t>0, (1.1) Lb,µ ≡vt−bvxx−µvx = +rv−suv ≡g(u,v), x ∈R, t>0, (1.2) supplemented with the initial conditions
u(x, 0) =u0(x), v(x, 0) =v0(x), x∈R. (1.3) The functionsu= u(x,t)andv=v(x,t)represent the densities of predators and preys in time t and at position x, respectively. The coefficient of diffusion a and b are positive constants which describe the rate of movement of predators and prey respectively. The nonnegative constants p andr are the coefficients of evolution, and the coefficientsq ands are related to the increase of the density of predators, and the decrease of the density of preys due to the presence of predators, respectively. The initial conditions u0 and v0 are two bounded and uniformly continuous functions onR.
BCorresponding author. Email: badraoui.salah@univ-guelma.dz
For a biomathematical discussion of these factors and for a background of the equations, see see [10] and [15].
For the modeling of this system see for example [12], page 53: if we have a lack where there are two species of fish: A, which lives on plants of which there is a plentiful supply, andB(the predator) which subsists by eating A(the prey), where u = u(x,t)represents the population ofBandv=v(x,t)represents the population of A.
Further, we suppose the domain is unbounded without boundary and no flux boundary con- ditions, instead of this we can suppose that the initial species distribution are describing by functions of finite supportu0andv0; namely, the initial conditions are of the form
u(x, 0) =u0(x) for −∆u< x<∆u, otherwiseu(x, 0) =0.
v(x, 0) =v0(x) for −∆v <x< ∆v, otherwisev(x, 0) =0.
where∆u and∆vgive the radius of the initially invaded domain, see [16].
The problem could be treated in the realistic two spatial dimension setting, in order to simplify the mathematics we are to treat it by one dimension space.
When the initial data are continuous, uniformly bounded, and nonnegative, it is shown that (1.1)–(1.2)–(1.3) has a classical positive global solution. Under some conditions on the coefficients or on the initial data, we show that this solution is in fact globally bounded.
Moreover, if
• r=0, p> qkv0k, thenvis bounded and u→0 exponentially ast→∞.
• p= 0,u0≥ k >r/sor u∗0 =min
u−0,u+0 >r/s, where u±0 = limx→±∞u0(x)then u(t) is bounded andv→0 exponentially ast →∞.
On the other hand, we study the behaviour of(u,v)whenx→ ±∞wheneveru0 andv0have limits at ±∞. We show that u(±∞,t) and v(±∞,t) satisfy an ordinary differential system (ODS) int. The qualitative behaviour of solutions to (1.1)–(1.2)–(1.3), asx → ±∞, can then be obtained from the ODS associated to it [7].
Some systems of predator–prey were studied in bounded domains, see [9,19] and in the references therein. Also, some results about global existence of solutions for systems of reaction-diffusion systems were established in [4,5,8,13,14].
In the following,u0andv0will be taken nonnegative and are elements of the Banach space X= (BUC(R),k·k), the space of bounded and uniformly continuous functions onRendowed with the supremum normkuk=supx∈R|u(x)|.
Note here that every continuous function of finite support is a uniformly continuous func- tion onR.
2 Existence, positivity and a priori bounds
We denote byA1 andA2 the linear operatorsa(·)xx+ν(·)x andb(·)xx+µ(·)x, respectively. It is well known thatAj, j=1, 2, generates an analytic semigroup of contractions on the Banach spaceXgiven explicitly by the expression
Sj(t)u
(x) = √ 1 4παt
Z +∞
−∞
"
exp −|x+σt−ξ|2 4αt
!#
u(ξ)dξ, (2.1)
whereα= aandσ =νforj=1, andα=bandσ=µforj=2.
Moreover, for any integern there is a positive constantc = c(n)such that for any u ∈ X, any positivetwe have DnSj(t)u∈ Xand the estimates
DnSj(t)u
≤ct−n/2kuk, (2.2)
where Dn=dn/(dx)n, andj=1, 2, holds true [6].
Our first result provides the existence of a global positive solution.
Theorem 2.1. Suppose that u0,v0 ∈ X,there exists a unique global classical nonnegative solution to the problem(1.1)–(1.2)–(1.3).
Proof. Local existence and uniqueness follow from standard arguments of abstract parabolic theory or from fixed point arguments involving the heat kernel and the Duhamel principle;
whence, there exists a t0 > 0 such that the problem (1.1)–(1.2)–(1.3) has a unique local mild solution(u,v)∈ C([0,t0];X)× C([0,t0];X), i.e.
u(t) =S1(t)u0+
Z t
0 S1(t−s)f(u(s),v(s))ds, t ∈[0,t0], v(t) =S2(t)v0+
Z t
0 S2(t−s)g(u(s),v(s))ds, t ∈[0,t0].
From the Lebesgue theory and the fact the functions (x,y)7−→ f(x,y)and(x,y) 7−→ g(x,y) are of classC∞(R2;R)we can conclude that the solution(u,v)∈ C∞(]0,T];X)× C∞(]0,T];X) for all 0 < T < Tmax, and (u(t),v(t)) ∈ C∞(R;R)× C∞(R;R)for allt ∈]0,T]; whereTmax is the maximal time of existence of the solution.
The continuous dependence of the solution on the initial data makes use of the local existence result and the Gronwall lemma.
The nonnegativity of the solution can be proved as follows: let λ1 = sup{ku(t)k, 0 ≤ t ≤ T}andλ2 = sup{kv(t)k, 0≤ t≤T}where 0 < T < Tmax(Tmax is the maximal time of existence of(u,v)), andλ0≥sup{r+sλ1,p+qλ2}. The substitutionsu=eλ0tϕandv=eλ0tψ transform system (1.1)–(1.2)–(1.3) into
ϕt−aϕxx−νϕx+ (p−qv+λ0)ϕ≡0, x∈ R, 0< t≤T, ψt−bψxx−µϕx+ (−r+su+λ0)ψ≡0, x∈ R, 0< t≤T, with
ϕ(x, 0) =e−λ0tu0(x)≥0 and ψ(x, 0) =e−λ0tv0(x)≥0, x ∈R.
Asu, v∈ C([0,T];X)and p−qv+λ0 ≥0 and−r+su+λ0 ≥0 for allt ∈[0,T], we can use Theorem 9 on page 43 of the maximum principle in [11] to get that ϕandψare nonnegative which in turn implies the nonnegativity ofuandv.
If one can establish the existence of a priori bounds for the solution components u, v on [0,Tmax[, standard continuation arguments yield global well posedness.
The solution to (1.1)–(1.2)–(1.3) can be written in integral form as follows u(t) =e−ptS1(t)u0+
Z t
0 e−p(t−τ)S1(t−τ)qu(τ)v(τ)dτ, (2.3) v(t) =e+rtS2(t)u0−
Z t
0
e+r(t−τ)S2(t−τ)su(τ)v(τ)dτ. (2.4)
Using the nonnegativity of(u,v)we get
kv(t)k ≤ertkv0k, for allt ≥0. (2.5) Using (2.3) and (2.5) we obtain
ku(t)k ≤ ku0k+qkv0k
Z t
0 erτku(τ)kdτ, for allt ≥0. (2.6) Gronwall’s inequality yields
ku(t)k ≤ ku0keqkv0kk(t), for all t≥0, (2.7) where
k(t) = (1
r ert−1
, ifr>0,
t, ifr=0.
Estimates (2.5) and (2.7) imply that the solution is global, i.e.,Tmax= +∞.
3 Boundedness and extinction results
The solution to (1.1)–(1.2)–(1.3) established in Theorem2.1is not always bounded as is shown in the following proposition.
Proposition 3.1. Assume v0 6≡0(v0 is not identically null) and r is sufficiently large, then(u,v)is unbounded. More precisely, v grows exponentially as t goes to∞.
Proof. Assume the contrary that (u,v) is a globally bounded solution, i.e., ku(t)k ≤ C and kv(t)k ≤ Cfor any t ≥ 0 and some constant C > 0. Asv0 6= 0, there exists a constantδ > 0 such thatS2(t)v0≥δ for anyt ≥0. Furthermore, we use (2.4) to obtain
v(t)≥ δ−sC2/r
ert+sC2/r, for allt ≥0.
Choosingr >sC2/δwe clearly havekv(t)k −→+∞astgoes to+∞. Whence(u,v)could not be bounded.
It is now clear that to get bounded solutions we have to impose some restrictions either on the coefficients of the system or on the initial data.
Theorem 3.2. If u0,v0∈ X then we have the estimates
kv(t)k ≤ kv0kert, for all t≥0, (3.1) ku(t)k ≤e(qkv0kerT−p)tku0k, for all t∈[0,T]. (3.2) Moreover, if r=0and p >qkv0kwe have
tlim→∞ku(t)k=0. (3.3)
Proof. Setting
u= ϕexp(−pt), (3.4)
v= ψexp(rt), (3.5)
the system (1.1)–(1.2) becomes
ϕt−aϕxx−νϕx= qertϕψ, (3.6)
ψt−bψxx−µψx= −se−ptϕψ, (3.7) with the initial data satisfying
ϕ0(x) =u0(x), (3.8)
ψ0(x) =v0(x). (3.9)
As ϕ≥0 andψ≥0, we first have from (3.7) and (3.9) ψ(t) =S2(t)v0−s
Z t
0 S2(t−τ)e−pτϕ(τ)ψ(τ)dτ≤S2(t)v0 ≤ kv0k, (3.10) for all (x,t)∈R×[0,T]. Whencev(t)≤ kv0kert, for allt ≥0.
Substituting (3.10) into (3.6) yields
ϕt−aϕxx−νϕx≤ qkv0kertϕ. (3.11) If we set ϕ=eMtw, where M =qkv0kerT, then we have overR×[0,T]
wt−awxx−νwx ≤0, w(x, 0) =ϕ0(x) =u0(x). (3.12) Furthermore
w(t) =S1(t)u0≤ ku0k, for allt≥0. (3.13) Whence ϕ≤eMtku0kand then
u(t)≤eMtku0ke−pt =e(qkv0kerT−p)tku0k, for all t∈[0,T]. Thus we obtain (3.2).
We deduce from (3.1)–(3.2) that ifr =0 andp >qkv0kwe will have kv(t)k ≤ kv0k, for all t≥0 and lim
t→∞ku(t)k=0.
Theorem 3.3. If r = 0, a ≤ b andν = µ, then the solution to (1.1)–(1.2) is globally bounded. We have the estimates
kv(t)k ≤ kv0k, for all t≥0, (3.14) and
ku(t)k ≤ ku0k+q s
√b/akv0k, for all t ≥0. (3.15)
Proof. LetYandZbe the solutions to
Yt−aYxx−νYx+pY=uv, Y(x, 0) =0, (3.16) and
Zt−bZxx−µZx =uv, Z(x, 0) =0, (3.17) respectively, where(u,v)is the solution to (1.1)–(1.2)–(1.3) withr =0, a≤ bandµ= ν. Then (u,v)can be written in terms of(Y,Z)as follows
u(x,t) =e−ptS1(t)u0(x) +qY(x,t), t≥0, (3.18) v(x,t) =S2(t)v0(x)−sZ(x,t), t ≥0. (3.19) Using the positivity of Z(x,t) we deduce (3.14) from (3.19). By the explicit formulas of Y andZ:
Y(t) =
Z t
0 e−p(t−τ)S1(t−τ)u(τ)v(τ)dτ≤
Z t
0S1(t−τ)u(τ)v(τ)dτ, (3.20) Z(t) =
Z t
0 S2(t−τ)u(τ)v(τ)dτ. (3.21) Asa ≤b,ν=µand (2.1), it is easy (see [1]) to deduce that
√aS1(t)w≤√
bS2(t)w, for allw∈X and then
S1(t)w≤ rb
aS2(t)w, for allt≥0. (3.22) From (3.20)–(3.22) we obtain
Y(t)≤ rb
a Z t
0 S2(t−τ)u(τ)v(τ)dτ= rb
aZ(t), for allt≥0. (3.23) Asvis nonnegative, from (3.19) we get
Z(x,t)≤ 1
sS2(t)v0, for allt≥0. (3.24) Using (3.24) in (3.23) we get
Y(t)≤ 1 s
rb
aS2(t)v0, for allt≥0. (3.25) Finally, from (3.25) in (3.18) we get (3.15).
Theorem 3.4. Assume p=0and u0≥r/s for all x ∈R.Then we have
kv(t)k ≤ kv0k, for all t≥0. (3.26) Moreover, if there is a constant k>r/s such that u0>k for all x ∈R, then
ku(t)k ≤
1+ q
ks−r ku0k
kv0k, for all t≥0, (3.27) and
kv(t)k ≤e−(ks−r)tkv0k, for all t≥0. (3.28) In particular, v−→0uniformly in x∈Ras t−→∞.
Proof. For p=0 andu0 ≥r/s, from (2.1) we get
u(t)≥r/s, for allt≥0. (3.29)
SettingB(t) =r−su(t), we have
vt = [A2+B(t)]v(t). (3.30) As the linear operator B(t) is dissipative on X [18], A2+B(t) generates for each t fixed a semigroup of contractions. Whence A2+B(t)generates on Xa system of evolutionP(t,τ)of contractions [18]. Whence the solution to (3.17)–(1.3) is
v(t) =P(t, 0)v0, for allt≥0. (3.31) This implies (3.26).
Ifu0 ≥ k> r/s, then from (1.1) we getu(t)≥k, and consequentlyr−su(t)≤ r−ks<0 for any t≥0. Settingω :=ks−r(ω >0), equation (1.2) can be written in the form
v(t) = [A2+B(t) +ωI]v(t)−ωv(t). (3.32) The dissipative operator B(t) +ωI generates on X a system of evolution G(t,τ) of contrac- tions. Consequently, A2+B(t)generates a system of evolutionU(t,τ)given by
U(t,τ) =e−ω(t−τ)G(t,τ). Hence the solution v(t)of (3.32)–(1.3) can be written in the form
v(t) =U(t, 0)v0=e−ωtG(t, 0)v0, for all t≥0. (3.33) This implies estimate (3.13). Using (1.1), (3.33) and Gronwall’s lemma we get (3.15).
In what follows, we denote byC±the closed subspaces ofXdefined as follows C± :=
u∈ Xsuch that : lim
x→±∞u(x)exists
.
Lemma 3.5. Let f ∈C±be such that f+, f−>0. Then for anyε>0there exists t∗ >0such that Sj(t)f
(x)≥ f∗−ε, for all x ∈R, where f∗ :=min(f+,f−).
Proof. The proof is similar to that of [4, Lemma 5.3].
In what follows we denoteu±0 =limx→±∞u0(x)andu∗0 =min
u−0,u+0 .
Theorem 3.6. Assume p =0and u0 ∈ C±. If u0∗ > r/s, then there exists t∗ > 0and three positive constants C1,C2andω∗ such that
kv(t)k ≤C1e−ω∗(t−t∗), for all t≥t∗, (3.34) ku(t)k ≤C2, for all t≥t∗. (3.35)
Proof. Chooseε > 0 such thatu∗0−ε > r/s, then by Lemma3.5, there existst∗ >0 such that [S1(t)u0] (x) ≥ u∗0−ε, for any x ∈ R. We then have u(t) ≥ u∗−ε, for any t ≥ t∗. Using Theorem3.4 with initial data(u(t∗),v(t∗))andk=u0∗−ε,ω∗ =ks−r, we then have
kv(t)k ≤ kv(t∗)ke−ω∗(t−t∗), for allt≥ t∗. We get (3.34) by setting C1 =kv(t∗)k.
Now, combining (2.3) and (3.34) we infer ku(t)k ≤ ku0k+qC1eω∗t∗
Z t
0
e−ω∗τku(τ)kdτ, for all t≥t∗. The Gronwall inequality yields
ku(t)k ≤ ku0keqCω∗1eω∗t∗ =C2, for allt ≥t∗. Whence (3.35).
4 Stability of the solution
Definition 4.1. We say that the solution to the problem (1.1)–(1.2)–(1.3) is unconditionally stable on R+, if for all T > 0 and all ε > 0, there exist δ = δ(T,ε) > 0 such that for all solution(u,v)with initial condition(u0,v0)to the same problem satisfyingku0−u0k<δand kv0−v0k<δ we haveku(t)−u(t)k<εandkv(t)−v(t)k<ε for allt∈ [0,T].
Proposition 4.2. The solution of the problem(1.1)–(1.2)is unconditionally stable onR+. Proof. From the integral writin of the solution(u,v)and(u,v)we get
ku(t)−u(t)k ≤ ku0−u0k+
Z t
0
{pku(τ)−u(τ)k+qku(τ)v(τ)−u(τ)v(t)k}dτ, (4.1) kv(t)−v(t)k ≤ kv0−v0k+
Z t
0
{rkv(τ)−v(τ)k+sku(τ)v(τ)−u(τ)v(t)k}dτ. (4.2) SettingΦ= (u,v),Φ= (u,v), Φ0= (u0,v0), Φ0= (u0,v0)and define
kΦ(t)k= k(u(t),v(t))k=ku(t)k+kv(t)k; then from (4.1)–(4.2) we get
Φ(t)−Φ(t)≤ Φ0−Φ0+ (p+r)
Z t
0
ku(τ)−u(τ)kdτ + (q+s)
Z t
0 ku(τ)v(τ)−u(τ)v(t)kdτ.
(4.3)
Letε>0 andT>0. Asu,v, u,v∈C(R+;X); then, they are bounded over[0,T]. Define kuk∞ = sup
t∈[0,T]
ku(t)k, for allu ∈ C R+;X
, (4.4)
then we have
kuv−uvk∞ ≤M
Φ(t)−Φ(t), for allt ∈[0,T], (4.5) where M=kuk∞+kvk∞.
From (4.3) and (4.5) we get
Φ(t)−Φ(t) ≤Φ0−Φ0+ [p+r+M(q+s)]
Z t
0
Φ(τ)−Φ(τ)dτ. (4.6) Using Gronwall inequality we obtain
Φ(t)−Φ(t)≤Φ0−Φ0e[p+r+M(q+s)]t, for allt ∈[0,T]. (4.7) The estimate (4.6) gives the stability of the solution to the problem (1.1)–(1.2)–(1.3).
5 Remarks
Remark 5.1. In turns out that ifu0,v0 ∈ C+then the diffusive system for x large will behave like the system of ordinary differential equations associated to it, and hence, for x large can be replaced by the latter which is simpler to analyze [7]
dU(t)
dt =−pU(t) +qU(t)V(t), for all t>0, dV(t)
dt = +rU(t)−sU(t)V(t), for all t>0, satisfying the initial data
U(0) = lim
x→+∞u0(x), V(0) = lim
x→+∞v0(x), where
U(t) = lim
x→+∞u(x,t), V(t) = lim
x→+∞u(x,t)
This result is based on the fact that ifh∈ C+withh+=limx→+∞h(x), then limx→+∞ Sj(t)h
(x)=
h+, forj=1, 2.
The same thing holds ifu0,v0∈ C−.
Remark 5.2. The same analysis can also be done for x ∈ [0,+∞[. In this case, the explicit formula associated to (1.1)–(1.2)–(1.3)
u(t) =e−ptS1(t)u0+
Z t
0e−p(t−τ)S1(t−τ)f(u(τ),v(τ))dτ, v(t) =e+rtS2(t)u0+
Z t
0 e+r(t−τ)S2(t−τ)g(u(τ),v(τ))dτ, will be
u(x,t) =
Z ∞
0
N1(x,ξ,t)u0(ξ)dξ+
Z t
0
x
t−τK1(x,t−τ)u1(τ)dτ +
Z t
0
Z +∞
0 N1(x,ξ,t−τ)f(u,v)(ξ,τ)dξdτ, and
v(x,t) =
Z ∞
0 N2(x,ξ,t)v0(ξ)dξ+ +
Z t
0
x
t−τK2(x,t−τ)v1(τ)dτ +
Z t
0
Z +∞
0
N2(x,ξ,t−τ)g(u,v)(ξ,τ)dξdτ,
where
N1(x,ξ,t) =K1(x−ξ,t)−K1(x+ξ,t), K1(x,t) = √ 1
4πatexp −|x+νt|2 4at
! , N2(x,ξ,t) =K2(x−ξ,t)−K2(x+ξ,t), K2(x,t) = √ 1
4πbtexp −|x+µt|2 4bt
! , and
u1(t) =u(0,t), v1(t) =v(0,t),
withu1,v1 bounded. These expressions can be deduced from [17, Chapter 3, Section 3].
It will be interesting to perform the same analysis for the case x ∈ [0,+∞[ with other boundary conditions.
Remark 5.3. For x∈Rn(n≥2)and replacingauxxandbvxxin (1.1)–(1.2) by the second order uniform elliptic operators
L1u=
∑
n i,j=1
aij(x)uxj
uxi, L2u=
∑
n i,j=1
bij(x)vxj vxi,
the problem deserves to be studied in appropriate functional spaces using the results in Aronson [2] and [3].
References
[1] W. F. Ames, C. Rogers,Nonlinear equations in the applied sciences, Mathematics in Science and Engineering, Vol. 185, Academic Press, Inc., Boston, MA, 1992.MR1132036
[2] D. G. Aronson, Nonnegative solutions of linear parabolic equations, Ann. Scuola Norm.
Sup. Pisa Cl. Sci. (3)22(1968), 607–694.MR0435594;Zbl 0182.13802
[3] D. G. Aronson, Bounds for the fundamental solution of a parabolic equation,Bull. Amer.
Math. Soc. 73(1967), 890–896. https://doi.org/10.1090/S0002-9904-1967-11830-5;
MR217444
[4] J. D. Avrin, Qualitative theory for a model of laminar flames with arbitrary nonneg- ative initial data, J. Differential Equations 84(1990), 290–308. https://doi.org/10.1016/
0022-0396(90)90080-9;MR1047571
[5] J. D. Avrin, M. Kirane, Temperature growth and temperatures bounds in special cases of combustion models, Appl. Anal. 50(1993), 131–144. https://doi.org/10.1080/
00036819308840189;MR1281208
[6] S. Badraoui, Z. Mehamedia, Uniform boundedness and decay estimates for a sys- tem of reaction-diffusion equations with a triangular diffusion matrix on Rn, Mediterr.
J. Math. 10(2013), 241–254. https://doi.org/10.1007/s00009-012-0239-8; MR3019103;
Zbl 1270.35254
[7] S. Badraoui, Asymptotic behavior of solutions to 2×2 reaction-diffusion system with a cross diffusion matrix on unbounded domains, Electron. J. Differential Equations 2006, No. 61, 1–13.MR2226934
[8] J. W. Bebernes, W. Fulks, The small heat-loss problem, J. Differential Equations 57(1985), 324–332.https://doi.org/10.1016/0022-0396(85)90059-2;MR0790279
[9] H. Cao, S. Fu, Global existence and convergence of solutions to a cross-diffusion cubic predator-prey system with stage structure for the prey,Bound. Value Probl. 2010, Art. ID 285961, 24 pp.https://doi.org/10.1155/2010/285961;MR2646370
[10] M. Farkas, Dynamical models in biology, Academic Press, Inc., San Diego, CA, 2001.
MR1836483
[11] A. Friedman, Partial differential equations of parabolic type, Robert E. Krieger Publisching Company Malabar, Florida, 1983.
[12] D. W. Jordan, P. Smith, Nonlinear differential equations, 4th Edition, Oxford University Press, 2007.MR2348881
[13] J. I. Kanel, On global initial boundary-value problems for reaction-diffusion systems with balance conditions, Nonlinear Anal. 37(1999), 971–995. https://doi.org/10.1016/
s0362-546x(97)00685-8;MR1689279
[14] B. Larrouturou, The equations of one-dimensional unsteady flame propagation: exis- tence and uniqueness,SIAM J. Math. Anal. 19(1988), 32–59. https://doi.org/10.1137/
0519003;MR0924543
[15] A. Okubo, S. A. Levin, Diffusion and ecological problems: mathematical models, Springer- Verlag, Berlin, 1980.MR572962
[16] S. Petrovskii, A. Morozov, B. L. Li, Regimes of biological invasion in a predator-prey system with the Allee effect, Bull. Math. Biol. 67(2005), 637–661. https://doi.org/10.
1016/j.bulm.2004.09.003;MR2215557.
[17] A. N. Tikhonov, A. A. Samarskii, Equations of mathematical physics, Dover Publications, 1990.MR1149383
[18] K. Yosida,Functional analysis, Springer-Verlag, Berlin-Heidelberg, 1965. MR0180824 [19] R. Zhang, L. Guo, S. Fu, Global behavior for a diffusive predator-prey model with stage
structure and nonlinear density restriction. II. The case in R1, Bound. Value Probl. 2009, Art. ID 654539, 19 pp.https://doi.org/10.1155/2009/654539;MR2570329.
